# Isometries of some classical function spaces among the composition operators

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Contemporary Mathematics Isometries of some classical function spaces among the composition operators María J. Martín and ragan Vukotić edicated to Professor Joseph Cima on the occasion of his 7th birthday Abstract. We give a simple and unified proof of the characterizations of all possible composition operators that are isometries of either a general Hardy space or a general weighted Bergman spaces of the disk. We do the same for the isometries of analytic Besov spaces (containing the irichlet space) among the composition operators with univalent symbols. Introduction Throughout this note, dm(θ) = (2π) 1 dθ will denote the normalized arc length measure on the unit circle T. We assume that the reader is familiar with the definition of the standard Hardy spaces H p of the disk (see [], for example). We write da for the normalized Lebesgue area measure on the unit disk : da(re iθ ) = π 1 rdrdθ. The weighted Bergman space A p w is the space of all L p (, w da) functions analytic in the disk, where w is a radial weight function: w(z) = w( z ), non-negative and integrable with respect to da. Every H p is a Banach space when 1 p <, and so is A p w when the weight w is reasonable (whenever the point evaluations are bounded; roughly speaking, w should not be zero too often near the unit circle). The unweighted Bergman space A p is obtained when w 1 (see [S] for the theory of these spaces); the standard weighted space A p α corresponds to the case w(z) = (α + 1)(1 z 2 ) α, 1 < α <. Given an analytic function ϕ in the unit disk such that ϕ(), the composition operator C ϕ with symbol ϕ defined by C ϕ f(z) = f(ϕ(z)) is always bounded on any H p or A p α space, in view of Littlewood s Subordination Theorem. The monographs [S1] and [CM] are standard sources for the theory of composition operators on such spaces. 2 Mathematics Subject Classification. Primary 47B33; Secondary 3H5. Key words and phrases. Hardy spaces, Bergman spaces, Besov spaces, Composition operators, Isometries. Both authors are supported by MCyT grant BFM C2-1 and also partially by MTM E (MEC Program Acciones Complementarias ), Spain. 1 c (copyright holder)

2 2 MARÍA J. MARTÍN AN RAGAN VUKOTIĆ Being a Hilbert space, the Hardy space H 2 has plenty of isometries. However, the only isometries of H 2 among the composition operators are the operators induced by inner functions that vanish at the origin. Nordgren ([N], p. 444) showed that if ϕ is inner and ϕ() = then C ϕ is an isometry of H 2 (alternatively, see p. 321 of [CM]). The converse follows, for example, from a result of Shapiro ([S2], p. 66). According to Cload [C], this characterization of isometries of H 2 among the composition operators had already been obtained in the unpublished thesis of Howard Schwarz [S]. Bayart [B] recently showed that every composition operator on H 2 which is similar to an isometry is induced by an inner function with a fixed point in the disk. The surjective isometries of the more general H p spaces have been described by Forelli [F] as weighted composition operators. A characterization of all isometries of H p does not seem to be known. In this note we prove that the only isometries (surjective or not) of H p, 1 p <, among the composition operators are again induced by inner functions that vanish at the origin (see Theorem 1.3 below). This fact may be known to some experts so our emphasis is on the method of proof, which also works for Bergman spaces. Kolaski [K] (see also [S], 2.8) gave a characterization of all surjective isometries of a weighted Bergman space A p α similar to that of Forelli s. Again, no characterization of all isometries of these spaces seems to be known. The (Hilbert) Bergman space A 2 α, of course, possesses plenty of isometries. In a recent preprint Carswell and Hammond [CH] have shown, among other results, that the only composition operators that are isometries of the weighted (Hilbert) Bergman space A 2 α are the rotations. We prove an analogous statement (Theorem 1.3) for an arbitrary space A p w with a radial weight, p 1, where Hilbert space methods no longer work. The surjective isometries of the Bloch space have been characterized in a well known work by Cima and Wogen [CW] while the surjective isometries of the general analytic Besov spaces B p and some related irichlet-type spaces have been described more recently by Hornor and Jamison [HJ]. Recall that an analytic function in the disk is said to belong to the space B p if its derivative is in the weighted Bergman space A p p 2. These spaces form an important scale of Möbiusinvariant spaces that includes the irichlet space (p = 2) and the Bloch space (as a limit case as p ). They have been studied by many authors (see [AFP], [Z], [GV] for some details). The isometries (not necessarily surjective) among the composition operators acting on the irichlet space B 2 have been characterized in [MV]. Here we describe all isometries of Besov spaces B p, 2 < p <, among the composition operators with univalent symbols (Theorem 1.4). The proof follows a similar pattern to that of the proofs for Hardy and Bergman spaces, with some variations typical of analytic Besov spaces. 1. Main results and their proofs 1.1. Hardy and Bergman spaces. We characterize all isometries among the composition operators on the general H p and A p spaces by giving an essentially unified proof. The crucial point in both statements is that the symbol ϕ of any isometry C ϕ must fix the origin. Once this has been established, we proceed using a simple idea that is probably known to some experts, at least in the Hilbert space context (p = 2). In order to prove the claim about fixing the origin, we first establish

3 ISOMETRIES AMONG THE COMPOSITION OPERATORS 3 an auxiliary result similar to several others that are often used in the theory of best approximation. Lemma 1.1. Let µ be a positive measure on the measure space, M a subspace of L p (, dµ), 1 p <, and let T be a linear isometry of M (not necessarily onto). Then T f T g p 2 T g dµ = f g p 2 g dµ for all f, g in the subspace M. Proof. We apply the standard method of variation of (differentiation with respect to) the parameter. Given two arbitrary functions f, g in M, define the function N f,g (t) = tf + g p dµ, t R. Then, as described in Theorem 2.6 of [LL] (with f and g permuted for our convenience), N f,g() = p g p 2 (gf + fg) dµ. 2 Since T is a linear isometry of M, we have N T f,t g (t) = N f,g (t). After evaluating the derivative of each side at t = we get g p 2 (gf + fg) dµ = T g p 2 (T gt f + T ft g) dµ. Since this holds for arbitrary f and g we may also replace g by ig. After a cancellation, this yields g p 2 (gf fg) dµ = T g p 2 (T gt f T ft g) dµ. Summing up the last two identities, we get g p 2 gf dµ = T g p 2 T gt f dµ, which implies the desired formula. From now on we assume that the weight w is not only radial but behaves reasonably well in the sense that A p w is a complete space. Proposition 1.2. If a composition operator C ϕ is an isometry (not necessarily onto) of either H p or A p w, 1 p <, then ϕ() =. Proof. Consider M = H p, a subspace of L p (T, dm), and M = A p w, a subspace of L p (, w da), respectively. Then set g 1 and T f = C ϕ f in Lemma 1.1 and use the standard reproducing property for the origin to get f(ϕ()) = C ϕ fdm = fdm = f() in the case of H p. Property to get 1 fwda = 2 T T For the weighted Bergman space A p w, use the Mean Value ( 2π ) f(re iθ )dm(θ) w(r)rdr = 2 1 f()w(r)rdr = c w f()

4 4 MARÍA J. MARTÍN AN RAGAN VUKOTIĆ (for some positive constant c w ) and, similarly, (f ϕ)wda = 2 1 ( 2π ) (f ϕ)(re iθ )dm(θ) w(r)rdr = c w f(ϕ()), hence also f(ϕ()) = f(). Finally, choose the identity map: f(z) z to deduce the statement in both cases. Proposition 1.2 could have been established by other methods but we decided to give preference to the application of Lemma 1.1 from approximation theory. Theorem 1.3. Let 1 p <. Then: (a) A composition operator C ϕ is an isometry of H p if and only if ϕ is inner and ϕ() =. (b) A composition operator C ϕ is an isometry of A p w if and only if ϕ is a rotation. Proof. (a) Since C ϕ is an isometry, we have z H p = ϕ H p, hence = z p H p ϕ p H p = T (1 ϕ p ) dm. Since ϕ 1 almost everywhere on T, it follows that 1 ϕ p = almost everywhere on T and, thus, ϕ is inner. We already know from the Corollary that ϕ() =. (b) In view of the Corollary (ϕ() = ) and the Schwarz Lemma, we get that ϕ(z) z for all z in. Since w is assumed to be a nontrivial weight, it must be strictly positive on a set of positive measure in, hence the equality = z p A p ϕ p w A = ( z p ϕ p ) w(z) da p w is only possible if ϕ(z) = z throughout, that is, when ϕ is a rotation Analytic Besov spaces. The analytic Besov space B p, 1 < p <, is defined as the set of all analytic functions in the disk such that f p B = p f() p + s p (f) = f() p + (p 1) f (z) p (1 z 2 ) p 2 da(z) <. These spaces are Banach spaces with the Möbius-invariant seminorm s p, in the sense that s p (f φ) = s p (f) for every disk automorphism φ. It is well known that B p B q when p < q. For the theory of B p and other conformally invariant spaces of analytic functions in the disk, we refer the reader to [AFP], [GV], and [Z], for example. A general composition operator on B p is not necessarily bounded, roughly speaking because too many points of the disk can be covered too many times by ϕ (for an exact condition for the boundedness in terms of the counting function, see [AFP]). However, the boundedness of C ϕ is guaranteed for every univalent symbol ϕ when p 2. Indeed, after applying the Schwarz-Pick Lemma and the change of

5 ISOMETRIES AMONG THE COMPOSITION OPERATORS 5 variable w = ϕ(z), we get s p p(f ϕ) = (p 1) f (ϕ(z)) p ϕ (z) p (1 z 2 ) p 2 da(z) (p 1) f (ϕ(z)) p (1 ϕ(z) 2 ) p 2 ϕ (z) 2 da(z) = (p 1) f (w) p (1 w 2 ) p 2 da(w) ϕ() (p 1) f (w) p (1 w 2 ) p 2 da(w), showing that s p (f ϕ) s p (f) for all f in B p, whenever ϕ is univalent and p 2. Theorem 1.4. Let ϕ be a univalent self-map of the disk and 2 < p <. Then the induced composition operator C ϕ is an isometry of B p if and only if ϕ is a rotation. Proof. The sufficiency of the condition is trivial. For the necessity, suppose C ϕ is an isometry of B p, 2 < p <. We first show that again we must have ϕ() =. Since s p (f ϕ) s p (f) in this case and f ϕ p B p = f(ϕ()) p + s p (f ϕ) = f() p + s p (f) = f p B p, it follows that f(ϕ()) f() for all f in B p. Writing ϕ() = a and choosing f to be the standard disk automorphism ϕ a (z) = a z 1 za that interchanges the origin and the point a, we get = ϕ a (a) = ϕ a (ϕ()) ϕ a () = a, hence ϕ() =. This proves the claim. Thus, if C ϕ is an isometry of B p, it must satisfy s p (f ϕ) = s p (f) for all f in B p. Choose f(z) z. Using the definition of s p, the Schwarz-Pick Lemma (note that p 2 > by assumption), and the change of variable ϕ(z) = w, we get = s p (z) p s p (ϕ) p = (p 1) (1 z 2 ) p 2 da(z) (p 1) ϕ (z) p (1 z 2 ) p 2 da(z) 1 (p 1) (1 ϕ(z) 2 ) p 2 ϕ (z) 2 da(z) = 1 (p 1) (1 w 2 ) p 2 da(w) ϕ() 1 (p 1) (1 w 2 ) p 2 da(w) =, hence equality must hold throughout. In particular, it must hold in the Schwarz- Pick Lemma (in the third line of the chain above) and so ϕ must be a disk automorphism. Since it also fixes the origin, it follows that ϕ is actually a rotation.

### arxiv:1409.7492v1 [math.cv] 26 Sep 2014

NECESSARY CONDITION FOR COMPACTNESS OF A DIFFERENCE OF COMPOSITION OPERATORS ON THE DIRICHLET SPACE MA LGORZATA MICHALSKA, ANDRZEJ M. MICHALSKI arxiv:409.749v [math.cv] 6 Sep 04 Abstract. Let ϕ be a self-map

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### In memory of Lars Hörmander

ON HÖRMANDER S SOLUTION OF THE -EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the -equation in the plane with a weight which permits growth near infinity carries

### PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

### PETER G. CASAZZA AND GITTA KUTYNIOK

A GENERALIZATION OF GRAM SCHMIDT ORTHOGONALIZATION GENERATING ALL PARSEVAL FRAMES PETER G. CASAZZA AND GITTA KUTYNIOK Abstract. Given an arbitrary finite sequence of vectors in a finite dimensional Hilbert

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### The Positive Supercyclicity Theorem

E extracta mathematicae Vol. 19, Núm. 1, 145 149 (2004) V Curso Espacios de Banach y Operadores. Laredo, Agosto de 2003. The Positive Supercyclicity Theorem F. León Saavedra Departamento de Matemáticas,

### A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 2015, 767 772 A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE José Manuel Enríquez-Salamanca and María José González

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh

Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp -. TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### On the structure of C -algebra generated by a family of partial isometries and multipliers

Armenian Journal of Mathematics Volume 7, Number 1, 2015, 50 58 On the structure of C -algebra generated by a family of partial isometries and multipliers A. Yu. Kuznetsova and Ye. V. Patrin Kazan Federal

### Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### 1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### On the Eigenvalues of Integral Operators

Çanaya Üniversitesi Fen-Edebiyat Faültesi, Journal of Arts and Sciences Say : 6 / Aral 006 On the Eigenvalues of Integral Operators Yüsel SOYKAN Abstract In this paper, we obtain asymptotic estimates of

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative

### MINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFT-INVARIANT SUBSPACES OF L 2 (R n )

MINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFT-INVARIANT SUBSPACES OF L 2 (R n ) MARCIN BOWNIK AND NORBERT KAIBLINGER Abstract. Let S be a shift-invariant subspace of L 2 (R n ) defined by N generators

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS

Publicacions Matemàtiques, Vol 42 (998), 53 63. A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt s A -weights satisfy a reverse Hölder inequality with an explicit

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### arxiv:0908.3127v2 [math.gt] 6 Sep 2009

MINIMAL SETS OF REIDEMEISTER MOVES arxiv:0908.3127v2 [math.gt] 6 Sep 2009 MICHAEL POLYAK Abstract. It is well known that any two diagrams representing the same oriented link are related by a finite sequence

### Asymptotic Behaviour and Cyclic Properties of Tree-shift Operators

Asymptotic Behaviour and Cyclic Properties of Tree-shift Operators University of Szeged, Bolyai Institute 29th July 2013, Gothenburg Banach Algebras and Applications 1 Introduction, Motivation Directed

### Note on some explicit formulae for twin prime counting function

Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### 2 Fourier Analysis and Analytic Functions

2 Fourier Analysis and Analytic Functions 2.1 Trigonometric Series One of the most important tools for the investigation of linear systems is Fourier analysis. Let f L 1 be a complex-valued Lebesgue-integrable

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### ON A GLOBALIZATION PROPERTY

APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### Monotone maps of R n are quasiconformal

Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### A new continuous dependence result for impulsive retarded functional differential equations

CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Math 504, Fall 2013 HW 3

Math 504, Fall 013 HW 3 1. Let F = F (x) be the field of rational functions over the field of order. Show that the extension K = F(x 1/6 ) of F is equal to F( x, x 1/3 ). Show that F(x 1/3 ) is separable

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr

### AN EXPANSION FORMULA FOR FRACTIONAL DERIVATIVES AND ITS APPLICATION. Abstract

AN EXPANSION FORMULA FOR FRACTIONAL DERIVATIVES AND ITS APPLICATION T. M. Atanackovic 1 and B. Stankovic 2 Dedicated to Professor Ivan H. Dimovski on the occasion of his 7th birthday Abstract An expansion

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### A Stronger Form of the Van den Berg-Kesten Inequality

A Stronger Form of the Van den Berg-Kesten Inequality Peter Winkler March 4, 2010 Abstract Let Q n := {0,1} n be the discrete hypercube with uniform probability distribution, and let A and B be two up-events

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### INVARIANT SUBSPACES ON MULTIPLY CONNECTED DOMAINS

Publicacions Matemàtiques, Vol 42 (1998), 521 557. INVARIANT SUBSPACES ON MULTIPLY CONNECTED DOMAINS Ali Abkar and Håkan Hedenmalm Abstract The lattice of invariant subspaces of several Banach spaces of

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### 0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup

456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### 1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1

Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between

### arxiv: v1 [math.ho] 7 Apr 2016

ON EXISTENCE OF A TRIANGLE WITH PRESCRIBED BISECTOR LENGTHS S. F. OSINKIN arxiv:1604.03794v1 [math.ho] 7 Apr 2016 Abstract. We suggest a geometric visualization of the process of constructing a triangle

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space

A new viewpoint on geometry of a lightlike hypersurface in a semi-euclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB97-0784

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

### Boundary value problems in complex analysis I

Boletín de la Asociación Matemática Venezolana, Vol. XII, No. (2005) 65 Boundary value problems in complex analysis I Heinrich Begehr Abstract A systematic investigation of basic boundary value problems

### arxiv:math/0202219v1 [math.co] 21 Feb 2002

RESTRICTED PERMUTATIONS BY PATTERNS OF TYPE (2, 1) arxiv:math/0202219v1 [math.co] 21 Feb 2002 TOUFIK MANSOUR LaBRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### COFINAL MAXIMAL CHAINS IN THE TURING DEGREES

COFINA MAXIMA CHAINS IN THE TURING DEGREES WEI WANG, IUZHEN WU, AND IANG YU Abstract. Assuming ZF C, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω 1 in the

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.